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A solvable nonic polynomial

Continuing from our demonstration that a certain sextic polynomial, which are not in general solvable, has an explicit factorization, we go on to describe how a class of degree 9 polynomials is solvable. Consider to be rational integers, not all zero, and the nonic polynomial

The claim is that all such polynomials are in fact solvable!

I will reveal the argument a little later, but it’ll be interesting to see what kind of arguments readers can come up with.